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Math Help - Inner Product Space

  1. #1
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    Inner Product Space

    Let V be an inner product space. Let \{e_n\}_1^\infty be a complete orthonormal sequence in V. Suppose that f,g \in V with f = \sum_{n=1}^{\infty}c_ne_n and g= \sum_{k=1}^\infty d_ke_k. Prove that \langle f, g \rangle = \sum_{n=1}^\infty c_n \bar{d_n}.

    My proof so far:

    \langle f, g \rangle = \langle \sum_{n=1}^{\infty}c_ne_n ,  \sum_{k=1}^\infty d_ke_k \rangle

    = \sum_{n=1}^{\infty}c_n \langle e_n ,  \sum_{k=1}^\infty d_ke_k \rangle

    = \sum_{n=1}^{\infty}c_n \bar{d_n} \langle e_n, e_n \rangle

    I definitely don't think I can pull that sum out in the first step because the e_n has the sum with it as well. I was thinking of trying the following:

    \langle \lim_{n\to \infty} \sum_{n=1}^N c_n e_n , \lim_{k \to \infty} \sum_{k=1}^K d_k e_k \rangle

    I just don't have any idea where to go from here. I think I can pull out the limits, but I don't know how that would help.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Benmath View Post
    Let V be an inner product space. Let \{e_n\}_1^\infty be a complete orthonormal sequence in V. Suppose that f,g \in V with f = \sum_{n=1}^{\infty}c_ne_n and g= \sum_{k=1}^\infty d_ke_k. Prove that \langle f, g \rangle = \sum_{n=1}^\infty c_n \bar{d_n}.

    My proof so far:

    \langle f, g \rangle = \langle \sum_{n=1}^{\infty}c_ne_n ,  \sum_{k=1}^\infty d_ke_k \rangle

    = \sum_{n=1}^{\infty}c_n \langle e_n ,  \sum_{k=1}^\infty d_ke_k \rangle

    = \sum_{n=1}^{\infty}c_n \bar{d_n} \langle e_n, e_n \rangle

    I definitely don't think I can pull that sum out in the first step because the e_n has the sum with it as well. I was thinking of trying the following:

    \langle \lim_{n\to \infty} \sum_{n=1}^N c_n e_n , \lim_{k \to \infty} \sum_{k=1}^K d_k e_k \rangle

    I just don't have any idea where to go from here. I think I can pull out the limits, but I don't know how that would help.
    What do you mean "pull out the limits"? Like \displaystyle \lim_{n\to\infty}\lim_{k\to\infty}\left\langle\sum  _{n=1}^{N}c_ne_n,\sum_{k=1}^{K}d_ke_k\right\rangle?
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  3. #3
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    I definitely think that I cannot just pull out the sum like I did in the top, so I was trying to use partial sums, but I don't know where to go from there. Yes, what I mean was pulling out the limits like you have, but I don't even know if that would be helpful.
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  4. #4
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    Opalg's Avatar
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    There is no problem about pulling the sums out of the inner products, because the inner product is a continuous function of its two arguments. If f = \sum_{n=1}^{\infty}c_ne_n then f is the limit of the finite sums \sum_{n=1}^{N}c_ne_n. It follows that

    \langle f, g \rangle = \lim_{N\to\infty}\Bigl\langle \sum_{n=1}^{N}c_ne_n, g \Bigr\rangle =  \lim_{N\to\infty}\sum_{n=1}^{N}c_n\langle e_n,g\rangle = \sum_{n=1}^{\infty}c_n\langle e_n,g\rangle,

    and similarly for the right-hand term in the inner product.
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